Page:Carroll - Euclid and His Modern Rivals.djvu/205

. § 4.] 'Thus the Theorem—if two angles are right angles, they are equal—has for its reciprocal—if two angles are equals, they are right angles.'

(This, by the way, is a capital instance of the distinction between 'technical' and 'logical.' Here the technical converse is wild nonsense, while the logical converse is of course as true as the Theorem itself: it is 'some cases of two angles being equal are cases of their being right.')

'All Propositions are direct, reciprocal, or contrary—allso closely connected that either of the two latter' (I presume he means 'the latter two') 'is a consequence of the other two.'

A 'consequence'! Can he mean a logical consequence? Would he let us make a syllogism of the three, using the 'direct' and 'reciprocal' (for instance) as premisses, and the 'contrary' as the conclusion?

However, let us first see what he means by a 'contrary' Proposition.

'It is a direct Proposition to prove that all points in a circle enjoy a certain property, e.g. the same distance from the centre.'

(This notion of sentient points, by the way, is very charming. I like to think of all the points in a circle really feeling a placid satisfaction in the thought that they are equidistant from the centre! They are infinite in number, and so can well afford to despise the arrogance of a point within, and to ignore the envious murmurs of a point without!)

'The contrary Proposition shows that all points taken outside or inside the figure do not enjoy this property.'